Triangles Perimeters

Task 207 ... Years 4 - 8

Summary

The concept of perimeter is placed in the unusual context of exploring polyiamonds. The task begins with the unit for measuring perimeter being defined as the side length of an equilateral triangle (moniamond). Then students are asked to create collections of tetriamonds, pentiamonds and hexiamonds and explore the various perimeters. The '-iamond' words aren't used, but teachers are encouraged to ask: Would you like to learn the mathematicians' words for shapes like that?. The challenge involves reasoning related to reducing the perimeter of a 16-iamond, by 'hiding' as many perimeters as possible of the unit triangles within the shape.

Triangle Perimeters also appears on the Picture Puzzles Shape & Measurement A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. It also introduces tessellations with '-iamonds'.

Materials

16 triangle tiles

Content

measurement, perimeter

reasoning

recording mathematics

sorting, classifying, ordering

spatial perception, 2D or 3D

tessellation

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Question 1 is asking for all the tetriamonds. The only ones other than the one shown are:

Perimeter = 6

Perimeter = 6

So, all three tetriamonds have a perimeter of 6.

Question 2 asks the students to make pentiamonds. If they do this in an ordered way, perhaps starting with the three tetriamonds and in each case taking a unit triangle for 'walk' around its perimeter - that is, trying every possible case - they will find that the unique pentiamonds are:

Perimeter = 7

Perimeter = 7

Perimeter = 7

Perimeter = 7

So, all four pentiamonds have a perimeter of 7.

Perhaps, based on this data, students will predict that there will be five hexiamonds, Question 3, and each will have a perimeter of 8. But, not so on either count. There are, in fact, 12 hexiamonds and they can be derived from the pentiamonds be taking a unit triangle for a walk in the same way as above. All except one have a perimeter of 8. Which one?

So now it is clear that we can't find a simple relationship between the size of the polyiamond and the perimeters all the examples of that size. But is it possible to predict the minimum and maximum perimeters for a given polyiamond?

An extension activity is to cut all the hexiamonds from triangle paper. Arranging them in different ways makes al sorts of interesting designs. For example, all 12 can be arranged to make a parallelogram with side lengths 4 and 9 or an isosceles trapezium with parallel sides of 7 and 11 and the other sides 4. Please send photos of your students successes with these. There are many ways to do each one.

The many ways to make the 16-iamond and it is suggested a class display board on which students can pin new discoveries over time. To make a 16-iamond, at least one side of every triangle must touch another triangle. In effect, this removes those two sides from the perimeter. So it seems reasonable that the 16-iamond with the longest perimeter would have the smallest number of touching pairs. Also that the shortest perimeter would have the largest number of touching pairs. As we have seen above there might be more than one way to do each of these.

So the display corner could be built around:

Current longest perimeter found.

Current shortest perimeter found.

Attempts to find 16-iamonds with perimeters between those two.

In a different direction, a challenge that links the problem to pattern and algebra is:

If I tell you any size '-iamond' and I tell you it is straight, can you tell me its perimeter?

Can you tell me the number of triangle sides touching inside it?

Can you check your answers another way?

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

The best material for a class lesson based around this task is Triangle Tiles or MiniGeofix triangles, both of which are available from Mathematics Centre. However if these are not available you could organise a lesser experience by using Triangle Paper, linked above.

The purpose of the lesson, as always, is to model some, or all of the Working Mathematically process. in this case highlights of the process are working with a colleague to:

break a problem into parts

try every possible case

collect data

make and test hypotheses (both numeric and spatial)

keep a journal of the investigation

publish a class display

At the same time you will be able to develop the concept of perimeter (and contrast it with area which can be measured by the number of unit triangles in an '-iamond') and develop spatial reasoning and perception. The card suggests the main stages of such a lesson.

At this stage, Triangle Perimeters does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.